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Full text of "Analytical Mechanics"

ENERGY
where co is the angular velocity of the body.    Therefore tl total kinetic energy of the rotating body is
T=f (mr2dm-i Jo
(IV)
where I is the moment of inertia of the body about the axis of rotation.
Comparing the expression for the kinetic energy of rotation with the expression for the kinetic energy of translation we observe that moment of inertia plays the same role in motion of rotation as mass, the linear inerti plays in motion of translation.
The expression for the kinetic energy of a rotating boc may be put in a little different form by substituting for its value in terms of the moment of inertia about a paral] axis through the center of mass. Thus
FIG. 101.
where vc is the velocity of the center of mass. We ha^ thus divided the kinetic energy into two parts  (a) kinet energy due to the motion of translation of the body as whole with the velocity of the center of mass, (b) kinet energy due to the rotation of the body about an axis throuj the center of mass.
156. Work Done in Increasing the Angular Velocity of a Rig Body.  It was shown in  154 that the work done again the kinetic reaction of a particle equals the increase in tl kinetic energy of the particle. Therefore the work doi against the kinetic reaction of any number of particles is t]